I am beginning to focus better on some subtle aspects of the expected
evolution of the general mathematical community.
The heart of the matter is that there will emerge an accepted new concept
of a complete answer to a mathematical question. At the moment, there are
two accepted notions of a definitive answer to a question:
1. A proof within ZFC.
2. A refutation within ZFC.
What will gradually emerge is a new accepted possibility:
3. A proof within ZFC of the equivalence of the question with a reflection
principle over ZFC + a large cardinal axiom. Most typically, this will be
the 1-consistency of a standard large cardinal axiom.
It will be gradually universally accepted that after doing 3, there is
nothing more to do in terms of "settling" the question. And since it is
accepted that nothing more can be done in terms of any issues of
provability, truth, determinate truth values, and the like, these three
will be the de facto standard by which "settling" is judged.
Also, large cardinals will continue to be the vehicle of choice for stating
alternative 3, thus assuring their permanent place in the history of
mathematics.
Furthermore, I predict that no additional principles for "settling"
statements of normal mathematics will be generally accepted and be
effectively used for normal mathematics during the 21st century. However,
for abnormal mathematics, we already know that when 1 and 2 are not
available, generally 3 is not either. Here I mean this in the strong sense
that when 1 and 2 are known not to be available, generally 3 is known not
to be available either. This situation will strengthen the already strong
move away from statements substantially higher up in logical complexity
than currently normal mathematics.
Under this prediction, there will continue to be absolutely no doubt as to
what standards are to be applied to determine whether a question has been
"settled", and will prevent the spectacle of people publishing papers
settling questions in opposite directions. That would be regarded as the
ultimate horror by the general mathematical community. Of course, this
setup only works for statements reasonably low down like normal
mathematics. The prospects for having universal acceptance by the general
mathematical community of any standard like this for statements higher up,
resulting in a similar guarantee of no settling in opposite directions,
seem very dim indeed.
In particular, in order to create such a standard higher up, one must
completely defeat rivals. E.g., must must defeat either V = L or very large
cardinals. The higher up one goes, the more alternatives have to be
defeated. But in the case at hand with 1 - 3 above, there does not appear
to be anything reasonable that suggests itself as an alternative that even
has to be defeated. That is a big difference. It is much easier to expect
big changes in the general math community when they have no decent
alternatives except just one that works very very well. And of course there
is the convenient point that the general mathematical community has moved
so sharply away from even the idea of going up in complexity for what they
regard as normal mathematics.
On the other hand, I made conjectures in posting 8:56AM 9/5/00 to the
effect that there will remain bounded arithmetic propositions with serious
logical difficulties which cannot be cured even with the entire large
cardinal hierarchy. But here I expect that during the 21st century, no one
will be able to do anything of any substance with these questions, so there
will be no practical issue regarding the evaluation of any proposed
"settling" or "determination of truth values." (Recall that it is the
question sin(n) > 0, where n is a very large natural, natural number).